Question

Consider a door that opens to only one side (as most doors do). A spring mechanism closes the door automatically. The state of the door at a given time $\mathit{t}$(measured in seconds) is determined by the angular displacement $\mathit{\theta }\left(t\right)$(measured in radians) and the angular velocity $\mathit{\omega }\left(t\right)\mathbf{=}\frac{\mathbf{d}\mathbf{\theta }}{\mathbf{d}\mathbf{t}}$. Note that is always positive or zero (since the door opens to only one side), but $\mathit{\omega }$can be positive or negative (depending on whether the door is opening or closing).

When the door is moving freely (nobody is pushing or pulling), its movement is subject to the following differential equations:

$\left|\begin{array}{c}\frac{d\theta }{dt}=\omega \\ \frac{d\omega }{dt}=-2\theta -3\omega \end{array}\right|$

(the definition of$\mathit{\omega }$) ($\mathbf{-}\mathbf{2}\mathit{\theta }$reflects the force of the spring, and $\mathbf{-}\mathbf{3}\mathit{\omega }$models friction).

1. Sketch a phase portrait for this system.
2. Discuss the movement of the door represented by the qualitatively different trajectories. For which initial states does the door slam (i.e., reach$\mathit{\theta }\mathbf{=}\mathbf{0}$with velocity $\mathit{\omega }\mathbf{<}\mathbf{0}$?

Short answer

(a) Sketch is in the explanation.

(b) all trajectories converge to origin and it observe that for condition$\omega <0$ door reaches$\theta <0$ .

Step-by-step solution

Calculate the Eigen values of given system.

Consider a door that opens to only one side. A spring mechanism closes the door automatically. The state of the door at a given time $t$is determined by the angular velocity $\theta \left(t\right)$and the angular velocity $\omega \left(t\right)=\frac{d\theta \left(t\right)}{dt}$.

When the door is moving freely, its movement is subject to the following differential equations.

$$\begin{gathered} \frac{\mathbf{d}\mathbf{\theta }}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{\omega } \\ \frac{\mathbf{d}\mathbf{\omega }}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\mathbf{2}\mathit{\theta }\mathbf{-}\mathbf{3}\mathit{\omega } \end{gathered}$$

Matrix A for the system is:

$A=\left[\begin{array}{cc}0& 1\\ -2& -3\end{array}\right]$

Eigen values are given by the characteristic equation $\left|A-\lambda I\right|=0$

Hence, the values of Eigen values are $\lambda =-1,-2$

The corresponding eigen vectors are:

(a) Sketch of a phase portrait of the system.

The phase portrait is as follows:

(b) Movement of door for different trajectory.

Let$E_{{\lambda }_1},E_{{\lambda }_2}$are Eigen spaces of${\lambda }_1$androle="math" localid="1664894365518" ${\lambda }_2$. Since both eigenvalues are negative.

Hence, the solution is:

$\overrightarrow{x}\left(t\right)=C_1{e}^{{\lambda }_{1}t}\overrightarrow{v_1}+C_2{e}^{{\lambda }_{2}t}\overrightarrow{v_2}$

Here, the values are$\overrightarrow{x}=\left[\begin{array}{c}\theta \\ \omega \end{array}\right]$,$\overrightarrow{v_1}=\left[\begin{array}{c}1\\ -1\end{array}\right]$corresponding torole="math" localid="1664894343401" ${\lambda }_1=-1$and $\overrightarrow{v_2}=\left[\begin{array}{c}1\\ -2\end{array}\right]$corresponding to${\lambda }_2=-2$.

As $t\to \infty ,\overrightarrow{x}\left(t\right)\to 0$solution converges to origin.

Also, $C_1$and $C_2$are arbitrary constants.

Dominant terms is $C_2{e}^{{\lambda }_{2}t}\overrightarrow{v_2}$as ${\lambda }_2$is larger in magnitude, therefore distant trajectories are parallel to $E_{{\lambda }_2}$.

All trajectories converge to origin.

Along the trajectory 1, the movement of the door slow down continuously until it reaches to the closed state as along this trajectory $w$decreases continuously.

Door slams if the trajectory 2 is followed that is initially if :

role="math" localid="1664894281284" $\frac{\omega \left(0\right)}{\theta \left(0\right)}<{\lambda }_2=-2$

From figure it observe that for condition $\omega <0$door reaches$\theta <0$ .

Hence, all trajectories converge to origin and it observe that for condition$\omega <0$ door reaches$\theta <0$ .